In the quarter circle above, what is the area of the shaded region?

From statement 1:

Perimeter of ABDF = 14 = 2(AB+AF)
Then AB+AF = 7.

No extra info. Hence insufficient.

From statement 2:

AC = 5 = radius of the quarter circle.
AF = 3.
AE = AC (Radius)
FE becomes 2.
No info about AB. Hence insufficient.

Combining both:

AB becomes 4.

Area of the quarter circle = \(\frac{90}{360}\)*Pi*5^2 = \(\frac{25}{4}\)Pi

Area of the shaded region = Area of the quarter circle-\(\frac{1}{2}\)Area of the rectangle
Area of the shaded region = \(\frac{25}{4}\)Pi-6

Hence sufficient.

C is the answer.

From statement 1: Perimeter of rectangle is known. With given data we will not be able to find the area of unshaded portion.
From statement 2: radious of quarter circle is known and base of unshaded portion is known. But,we will not be able to find the area of unshaded portion.

Combining both the statements, area of unshaded portion and quarter circle can be found out.

Hence, option C is correct. IMO.

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In the quarter circle above, what is the area of the shaded region?

(1) The perimeter of the rectangle ABDF is 14.
(2) AC is 5 and AF is 3

(1)--> 2(l+w)=14 --> (l+w)=7 -->clearly insuff. (l and w values not given)
(2)--> AC=5=radius and AF=l=3 --> insuff. (we don't know what value can b can take)

Combining (1) and (2),
(1)--> l+w=7 and (2)-->l=3, we can find b=4;
Since rectangle forms a right angle,
Area of unshaded=1/2 * l * b -->Find the area of BAF
Area of circle = 25pi -->(r=5)

Hence, Area of shaded=Area of circle-area of unshaded -->Suff

Ans C

Bunuel, How's the answer B and not C?

Hello, How can AD could be 5?

A is the center of the quarter circle. And AD, AE, and AC are all radii. Hence AD is 5.

Area of shaded region = area of quarter of the circle - area of triangle

Translated into formulas, we get: \((1/4)pi*r^2 - (1/2) base*height\)

Thus, the unknowns we need to solve the question are r, base, and height.

(1) This statement does not give us all the unknowns we need. Insufficient
(2) This statement provides us with the radius (AC =5) and the base (AF=3). Now, we also know that the diagonal cuts the rectangle in half and therefore two of its angles are halfed into 45°, making the resulting two triangles into two identical isosceles triangles. Knowing that the ratio of the sides of a right isosceles triangle is x:x:x\(\sqrt{2}\), we also know that base (AC) = height (AB) =5. Based on this information, we can calculate the area of the quarter circle and the area of the triangle.

Statement (2) is sufficient. Answer B

Please hit Kudos if you liked this answer.

I understand the logic of B, but how can you guarantee ABDF is a rectangle? if BD is not parallel to AF (so assume AB=3 and DF=4), then the diagonals are not equal, leading to C as the correct answer. Can someone please explain how 2) ensures that ABDF is a rectangle on its own?

Trick here is AC=AD=5, since both are radius.

Now everybody agree that statement 1 is not sufficient. Lets directly move to statement 2.
Here AC=AD=5, AF=3,

By using Pythagoras theorem AB=DF=4

Finally area of shaded reason= area of quarter circle-(Half area of rectangle).
Hence required area can be easily calculated since we know radius and both sides of rectangle.


Statement 2 is sufficient.

BTW i got it wrong too.

Shaded area= pi* r^2- 1/2(area of rectangle)
Also, radius of quarter circle, r= Diagonal of the rectangle

We need the 2 sides of rectangle to find the area of shaded area

Statement 1- We have the sum of the sides of rectangle.
Insufficient

Statement 2- we know the radius or the diagonal of rectangle, and one of the side of rectangle; Hence, we can figure out the second side of rectangle, as well as the area of the shaded portion.

Sufficient.

Statement 2: How do you know the polygon is a rectangle?